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Properties

Label 1-155-155.119-r1-0-0
Degree $1$
Conductor $155$
Sign $-0.970 + 0.242i$
Analytic cond. $16.6570$
Root an. cond. $16.6570$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−0.5 − 0.866i)3-s + 4-s + (0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + 16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  − 2-s + (−0.5 − 0.866i)3-s + 4-s + (0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + 16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $-0.970 + 0.242i$
Analytic conductor: \(16.6570\)
Root analytic conductor: \(16.6570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 155,\ (1:\ ),\ -0.970 + 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01767066748 - 0.1435588858i\)
\(L(\frac12)\) \(\approx\) \(0.01767066748 - 0.1435588858i\)
\(L(1)\) \(\approx\) \(0.4896129383 - 0.1223872502i\)
\(L(1)\) \(\approx\) \(0.4896129383 - 0.1223872502i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 - T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 - T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 - T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.8232695619183127359908883898, −27.44111080700430343046056734818, −26.526518890916080249597254337707, −25.63513270706244931934401886109, −24.47575599172676641658367834379, −23.337725686173449880471569877068, −22.3336053943155218181060131882, −21.01389262728294192591994839393, −20.39353619591355035282021115879, −19.46448386507716592503119857105, −17.933845155318328538866740290035, −17.19402364296213335810834771957, −16.6796166447311155165011483415, −15.19801307303758749387555270048, −14.7541902158780116599534818480, −12.70384447746334613147586912978, −11.51588418640022839884351406754, −10.53677037023730347314729631970, −9.94047416009955340205559393280, −8.69000229484979972088696504207, −7.46862921420706950891721268645, −6.29346764548184932925896330579, −4.83187362584351576154909521928, −3.4939889359400523754174190907, −1.567149635087215135043092638293, 0.0798949401602426501760589280, 1.59828761187733082135311347383, 2.701232499586324890538414187519, 5.13791095361353085149090383051, 6.388381265930823331464719819173, 7.24196598428199884356517014995, 8.53936939193888013458094024912, 9.27750606110841811179779302278, 11.167117170706126848051207464209, 11.45898759220388558917279462089, 12.62286989043637345395936357156, 14.06447007030753605250233844961, 15.323112894793993940605656822069, 16.582526938262795034786723962077, 17.2996818111625762438219250625, 18.365850186439648837459673687401, 18.964627807392593605133546469425, 19.84357592411143532745253306673, 21.28357429261312839022234629309, 22.13689661576410154572516909538, 23.684128279077905610801057729244, 24.56850373430726322588121042420, 24.96854372174220847130537678145, 26.295138960074490217990675156296, 27.334501094755489338758626022042

Graph of the $Z$-function along the critical line